Let S be the set of points, whose abscissae and ordinates are natural numbers. Let p e S, such that the sum of the distance of P from (8, 0) and (0, 12) is minimum among all elemants in S. Then, the number of such points P in S is

• 1

• 3

• 5

• 11

If in a $∆$ABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circumcircle of $∆$DEF is

• $\frac{\mathrm{R}}{2}$

• $\frac{2\mathrm{R}}{3}$

• $\frac{\mathrm{R}}{3}$

• None of these

The line AB cuts off equal intercepts 2a from the axes. From any point P on the line AB perpendiculars PR and PS are drawn on the axes. Locus of mid-point of RS is

• $\mathrm{x} - \mathrm{y} = \frac{\mathrm{a}}{2}$

• x + y = a

• x² + y² = 4a²

• x² - y² = 2a²

X + 8y - 22 = 0, 5x + 2y -34 = 0, 2x - 3y + 13= 0 are the three sides of a triangle. The area of the triangle is

• 36 sq units

• 19 sq units

• 42 sq units

• 72 sq units

The line through the points (a, b) and (- a,- b) passes through the point

• (1, 1)

• (3a, - 2b)

• (a², ab)

• (a, b)

The locus of the point of intersection of the straight  lines $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} = \mathrm{K} \mathrm{and} \frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} = \frac{1}{\mathrm{K}}$ where K is a non-zero real variable, is given by

• a straight line

• an ellipse

• a parabola

• a hyperbola

37.

The equation x³ - yx² + x - y = 0 represents

• a hyperbola and two straight lines

• a straight line

• a parabola and two straight lines

• a straight line and a circle

The coordinates of a point on the line x + y + 1 = 0, which is at a distance $\frac{1}{5}$ unit from the line 3x + 4y + 2 = 0, are

• (2, - 3)

• (- 3, 2)

• (0, - 1)

• (- 1, 0)

The least positive value of t, so that the lines $\mathrm{x} = \mathrm{t} + \mathrm{\alpha }, \mathrm{y} + 16 = 0 \mathrm{and} \mathrm{y} = \mathrm{\alpha x}$ are concurrent is

• 2

• 4

• 16

• 8

Number of points having distance $\sqrt{5}$  from the straight line x - 2y + 1 = 0 and a distance $\sqrt{13}$ is from the line 2x + 3y - 1 = 0, is

• 1

• 2

• 4

• 5